Monte Carlo Renormalization Group for Systems with Quenched Disorder

We extend to quenched disordered systems the variational scheme for real space renormalization
group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder
is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from
which one can compute the critical exponents if the correlations of the renormalized couplings retain
finite range. Key to the variational approach is the bias potential found by minimizing a convex
functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation
time in large disordered systems. We demonstrate the method with applications to dilute Ising,
random field Ising and short-range spin glass models, on the two-dimensional square lattice.